Re: [理工] 工數 配方法問題?
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部分分式用heaviside cover-up method
就是所謂的手遮帶值法
1.
3 3 A B Cs+D
------------------ = ----------------- = --- + --- + -----
(s^2+9)(s^2+7s+12) (s^2+9)(s+3)(s+4) s+3 s+4 s^2+9
令F(s)=3/[(s^2+9)(s+3)(s+4)]
| 3 1
A = F(s)(s+3)| = --- = ---
|s=-3 18 6
| -3
B = F(s)(s+4)| = ---
|s=-4 25
| 3 3(7s-3) 21s-9 21s-9 7s-3
Cs+D = F(s)(s^2+9)| = ------ = ------------ = ------- = ----- = -----
|s^2=-9 (7s+3) (7s+3)(7s-3) 49s^2-9 -450 -150
=> C = -7/150, D = 3/150 = 1/50
A的部分解答錯了
2.
64(s+1) A B C D
-------- = --- + --- + --- + ---
s^3(s-4) s-4 s^3 s^2 s
令F(s)=[64(s+1)]/[s^3(s-4)]
| 64*5
A = F(s)(s-4)| = ---- = 5
|s=4 4^3
| 64
B = F(s)s^3| = ---- = -16
|s=0 -4
d | 64[(s-4)-(s+1)]| 64*(-5)
C = --- (F(s)s^3)| = ---------------| = ------- = -20
ds |s=0 (s-4)^2 |s=0 16
d^2 | 320 | 320
D = ---- (0.5F(s)s^3)| = -------| = --- = -5
ds^2 |s=0 (s-4)^3|s=0 -64
說明一下重根公式的原理
64(s+1) As^3
F(s)s^3 = ------- = ---- + B + Cs + Ds^2
s-4 s-4
此時代s=0可直接求B
微分一次之後B不見了 A,D都還有s一次以上 此時代0可求C
微分兩次之後C也不見了 A還有s一次 Ds^2微兩次變2D
故此時代0可得2D 所以記得要除以2
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當然重根次數高就要微更多次
此題求C,D要微分可能有人已經覺得不太方便
D還有一個求法是取極限
F(s)s = 0 = A + D => D = -A = -5
s->∞
此時再隨便帶一個不會讓分母為0的值 例如代s=-1
=> 0 = -A/5 - B + C - D
C = A/5 + B + D = 1-16-5 = -20
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11/29 22:04, , 1F
11/29 22:04, 1F
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